A small business is launching a new social media marketing campaign and needs to set realistic engagement goals and analyze their actual results. The marketing team creates a plan to steadily increase their weekly engagement. They model their engagement goals using the explicit formula:

[latex]a_n = 500 + 150n[/latex]

where [latex]a_n[/latex] represents their target engagement (likes, comments, shares) in week [latex]n[/latex].

After running the campaign for several weeks, the marketing team collects their actual engagement data.

• Week 1: 800 engagements
• Week 2: 1,200 engagements
• Week 3: 1,800 engagements
• Week 4: 2,700 engagements
• Week 5: 4,050 engagements

The marketing team notices that their engagement growth is accelerating due to viral sharing effects. They predict that future weekly engagement will follow this pattern:

• Start with 4,050 engagements (their week 5 result)
• Each week, engagement will be 120% of the previous week plus an additional 300 engagements from new followers